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Causality
Pierre Grange, Ernst Werner
To cite this version:
hal-00118075, version 2 - 8 Feb 2007
Pierre GRANG´E
Laboratoire de Physique Th´eorique et Astroparticules Universit´e Montpellier II, CNRS, IN2P3
CC 070 - Bat.13 - Place E. Bataillon F-34095, Montpellier, Cedex 05 - FRANCE
Ernst WERNER
Institut f¨ur Theoretische Physik D-93040, Universit¨at Regensburg - GERMANY
(Dated: February 8, 2007)
Quantum Field Theory with fields as Operator Valued Distributions with adequate test functions, -the basis of Epstein-Glaser approach known now as Causal Perturbation Theory-, is recalled. Its recent revival is due to new developments in understanding its renormalization structure, which was a major and somehow fatal disease to its widespread use in the seventies. In keeping with the usual way of definition of integrals of differential forms, fields are defined through integrals over the whole manifold, which are given an atlas-independent meaning with the help of the partition of unity. Using such partition of unity test functions turns out to be the key to the fulfilment of the Poincar´e commutator algebra as well as to provide a direct Lorentz invariant scheme to the Epstein-Glaser extension procedure of singular distributions. These test functions also simplify the analysis of QFT behaviour both in the UV and IR domains, leaving only a finite renormalization at a point related to the arbitrary scale present in the test functions. Some well known UV and IR cases are examplified. Finally the possible implementation of Epstein-Glaser approach in light-front field theory is discussed, focussing on the intrinsic non-pertubative character of the initial light-cone interaction Hamiltonian and on the expected benefits of a divergence-free procedure with only finite RG-analysis on physical observables in the end.
PACS numbers: 11.10.Ef, 11.10.St, 11.30.Rd
I. INTRODUCTION
Causal perturbation theory (CPT) goes back to ideas from Stueckelberg, Rivier and Green [1], Bogoliubov, Parasiuk and Sirkov [2] and has been developed by Epstein and Glaser [3] and more recently by Scharf [4]. A little after Epstein and Glaser came formulations [5] to construct finite gauge theories, based on regulator free momentum substraction schemes. In the last publication the questions of the finiteness of Lorentz-invariant amplitudes and of symmetry conservation are treated seperately. Moreover, as opposed to CPT, the importance of causality issues in relation to the finiteness problem itself is not of main concern. In CPT the S-matrix is constructed as a formal functional expansion in an unimodular interaction-switching test function. Whereas in the traditional approach the S-matrix amplitudes are determined by equations of motion, in CPT they are determined inductively by causality conditions imposed on switching test functions. Such causality conditions turn out to be as stringent for the S-matrix evolution as the equations of motion themselves. On an heuristic level it may be seen as follows: if g1(t) and g2(t) are two such switching test
functions with supports such that supp[g1] ∈ (−∞, s)
and supp[g2] ∈ (s, +∞) respectively, then causality
implies that S(g1 + g2) = S(g1)· S(g2), which is the
functional equation for an exponential.
The S-matrix amplitudes are time ordered products of operator valued distributions (OPVD). They are split into causal advanced and retarded pieces, in such a way that any singular behaviour at equal space-time points is avoided. The procedure has long been recognized as mathematically rigorous [6, 7] since it is divergence free at each iteration stage. Nevertheless Epstein-Glaser work almost fell into oblivion among QFT practioners mainly because
(1) in its original version there were difficulties in disen-tangling the multiplicative structure of renormalization and
(2) the rise of renormalization group methods turned research interests into this direction.
in-cludes BPHZ renormalization [2, 11] as a special case. In a Minkowskian metric this is equivalent to the implemen-tation of causality while in its Euclidean counterpart it is a symmetry preserving prescription for substractions. In light-front quantum field theory (LFQFT) there is a compelling reason to introduce test functions related to the consistency of the canonical quantization scheme it-self. It is best seen for the massive scalar field. The form of the LF-Laplace operator leads to a hyperbolic equation of motion which requires initial data on two characteristics. Canonical quantization in terms of ini-tial field values in the light cone time is however possible provided [12] limp+→0χ(p
+)
p+ = 0,where χ(p+) is the field
amplitude at p+ = p0+ p3. With fields as OPVD this
relation becomes limp+→0f (p +)
p+ , which is surely satisfied
for the class of test functions f (p+) used for the Fock
expansion of the field operators [13, 14].
Our earlier approach with test functions [13, 16] in fact implements Epstein-Glaser treatment of singular distri-butions in the LFQFT context. Here we want to as-sess our particular choice of test functions as ‘partition of unity‘ and show the resulting simplifications brought about for the treatment of Poincar´e’s invariance and for the Lorentz invariant Taylor series surgeries discussed in [10].
In Section II the general definition of fields as OPVD is given. Due to specific properties of Euclidean manifolds it is shown that the class of admissible test functions can be restricted to a ‘partition of unity‘, in close analogy with the usual atlas-independent definition of integrals of differential forms. Thus a QFT construction is possible, which is test-function independent. As a consequence, for scalar field theory it is shown that Poincar´e’s com-mutator algebra is fulfilled, in contrast with the case of arbitrary test-functions. In Section III the Bogoliubov-Epstein-Glaser expansion of the S-matrix is recalled. It is given in terms of the interaction-switching test function g ∈ D′(IR 4). The proper Epstein-Glaser time ordered
construction of the expansion coefficients is described. The implementation of Lorentz covariance of the proce-dure is presented and the simplifications brought about with test functions vanishing at the origin with all their derivatives are shown. They concern Lorentz covariance, the Taylor surgery itself and its interpretation in terms of the BPHZ renormalisation scheme. In Section IV, the use of Lagrange’s formula [8, 10] for Taylor’s remainder in connection with partition of unity test functions is shown to simplify also the analysis of singular distributions both in the IR and UV domains. A direct application of the method in the IR is shown to be successful for the free massive scalar field theory developed in the (convention-ally hopeless) mass perturbation expansion. It should also prove relevant to many problems of massless confor-mal field theories (CFT). Finally, following [17], in the UV the alternative interpretation of the effects of parti-tion of unity test funcparti-tions in terms of Pauli-Villars type subtractions at the level of propagators is pointed out
in Euclidean and Minskowskian metrics. In Section V Epstein-Glaser separation method of Section III for dis-tributions into advanced and retarded pieces (splitting procedure) is rephrased for the special case of partition of unity test functions. Their use simplifies greatly the derivation of the relation between the splitting proce-dure and dispersion relations. In Section VI a general discussion is given on the possibility of applying BSEG’s method in a non-perturbative Light Front Quantization framework and on the benefits one might expect from the procedure. Finally some conclusions are drawn in Section VII.
II. FIELDS AS OPVD, PARTITION OF UNITY
AND POINCAR´E COMMUTATOR ALGEBRA
A. Fields as OPVD
The history of fields as OPVD is almost as old as Quan-tum Field Theory itself. In fact the mathematical devel-opments of Distribution Theory (or Generalized Func-tions) [18, 20] in the 1960′s were intimately related to
problems raised at that time by pioneers in the QFT for-mulation. This mutual feedback has lead to QFT formu-lations [21] in the 1960−70′s which aimed at dealing in a
mathematically consistent way with intrinsic divergences crippling the usual QFT perturbative analysis. Among these approaches the work of Epstein and Glaser [3] has long been recognized as mathematically rigourous and free of undefined quantities. It is still generating an im-portant literature in mathematical physics and we shall summerise here some of its recent developments.
To introduce fields as OPVD one may consider, with-out loss of generality, the free massive scalar field in D-dimensions. The general solution of the Klein-Gordon equation is a distribution, ie an OPVD, which defines a functional with respect to a test function ρ(x) belonging toD(IR D), the space of Schwartz test functions [18],
Φ(ρ)≡< φ, ρ >= Z
d(D)yφ(y)ρ(y). (II.1)
Here Φ(ρ) is an operator-valued functional with the pos-sible interpretation of a more general functional Φ(x, ρ) evaluated at x = 0. Indeed the translated functional is a well defined object [18] such that
TxΦ(ρ) = < Txφ, ρ >=< φ, T−xρ > (II.2)
= Z
d(D)yφ(y)ρ(x− y). (II.3)
It follows that TxΦ(ρ) = Z d(D)p (2π)De −ipxδ(p2 − m2)χ(p)f (p20, p2).
Due to the properties of ρ, TxΦ(ρ) obeys the KG-equation
and is taken as the physical field ϕ(x). After integration over p0 and with ωp2 = p2+ m2 the quantized form is
taken as the (D− 1)−Euclidean integral
ϕ(x) = Z d(D−1)p (2π)(D−1) f (ω2 p, p2) 2ωp
[a+peipx+ ape−ipx]. (II.5)
f (ω2
p, p2) acts as a regulator [13, 14] with very specific
properties [15]. This expression for ϕ(x) is particularly useful to define correlation functions of the field and more precisely in the light-cone (LC) formalism because the Haag series can be used [16] and is well defined in terms of the product of ϕ(xi). In the full Euclidean
metric there is no on-shell condition and ϕ(x) stays a D-dimensional Fourier transform with f (p2) only.
It might appear that there would be as many QFT’s as eligible test functions. However some importants results from topological spaces analysis may be invoked. Some general definitions need first to be recalled :
Definitions:
a) An open covering of a topological space M is a family of open sets (Ui)i∈I such thatM = ∪i∈IUi.
b) An open covering (Ui)i∈I is said to be locally finite
if each point ofM has a neighbourhood which intersects only a finite number of the (Ui).
c) A topological space M is paracompact iff every open covering (Ui)i∈I admits a locally finite refinement
(that is a locally finite open covering (Vj)j∈J with the
property that for all j ∈ J there exists i ∈ I such that Vj⊆ Ui).
d) If (Ui)i∈I is an open covering of M, a partition
of unity subordinate to (Ui)i∈I is a family (βj)j∈J of
positive continuous functions onM such that
(i) For all X ∈ M there exists a neighbourhood UX of X such that all but a finite number of the βj
vanish on UX.
(ii)Pi∈Iβi(X) = 1 for all X∈ M.
(iii) For all j ∈ J, there exists i ∈ I such that the closure of{X ∈ M : βj(X)6= 0} is contained in Ui. The
closure of{X ∈ M : βj(X)6= 0} is called the support of
βj.
Remark: If (αi)i∈I and (βj)j∈J are two partitions of
unity subordinate to (Ui)i∈I and (Vj)j∈J respectively,
(αiβj)(i,j)∈I×J is a partition of unity subordinate to
(Ui∩ Vj)(i,j)∈I×J. Since this property is crucial in the
sequel it is established in Appendix A. Then the following theorems hold: Theorems:
A) Suppose M connected. Then M is paracompact iff the topology of M is metrisable iff (when M is a differentiable manifold)M admits a Riemannian metric. B) If M is a connected, paracompact differentiable manifold and (Ui) is an open covering, then there
ex-ists a differentiable partition of unity subordinate to (Ui).
Partitions of unity are useful because one can often use them to extend local constructions to the whole space. A well known case deals with the definition of integrals of differential forms [22]: let F be a differential form, (βi) is used to cut F into small pieces: Fi = βiF and
P
Fi= F . For Ω⊂ M one defines:
R ΩF := P i R ΩiβiF .
βi being of compact support each integral in the sum is
finite and the overall result is independent of the choice of coordinates (atlas) on Ωiand of the partition of unity.
C) Localisation of distributions: J. Dieudonn´e’s GPT-theorem (Glueing-Pieces-Together)[18, 19] establishes the above properties for distribution functionals.
The metrisable Euclidean manifold being paracompact permits the use of partition of unity test functions in the definition of the field of Eq.(II.5). The resulting operator-valued functional is independent of the way this partition of unity is constructed. All constructions of the field with different partition of unity are therefore equivalent, thereby eliminating the initial arbitrariness in the choice of test functions. Then f (p2) is 1 execpt in the
bound-ary region where it is infinitely differentiable and goes to zero with all its derivatives. An immediate and impor-tant consequence of this construction is that all successive convolutions of the field ϕ(x) with the test function leave ϕ(x) unchanged. One has
< Txϕ, ρ >= Z d(D−1)p (2π)(D−1) f2(ω2 p, p2) 2ωp [a+peipx+ ape−ipx]. (II.6) Here f2stands for the product of two partitions of unity.
According to the Remark above and Appendix A f2 is also a partition of unity with the same support as f and ϕ(x) is not affected.
B. Partition of unity and Poincar´e commutator algebra
and use the following conventions: dΩk= d 3k (2π)32ω k; ωk= √ k2+ m2; [a ~ k, a + ~ k′] = (2π) 32ω kδ(~k − ~k′). The
energy-momentum tensor writes θµν = ∂µϕ∂νϕ
−12g µν[(∂ϕ)2 − m2ϕ2] and gives P0= H = Z dΩkωka+~ ka~kf 2(ω2 k, ~k2); Pj = 1 2 Z dΩkωkkj[a~+ ka~k+ a~ka + ~ k]f 2(ω2 k, ~k2); (II.7) M0j= i Z dΩkωka~+kf (ω 2 k, ~k2) ∂ ∂kj(a~kf (ω 2 k, ~k2)); Mjl= i Z dΩka~+kf (ω 2 k, ~k2)[kj ∂ ∂kl − kl ∂ ∂kj](a~kf (ω 2 k, ~k2)). (II.8)
Evaluating the commutator of Pjwith ϕ(x) gives in turn
i[Pj, ϕ(x)] = i Z dΩkkj[a~+ ke ikx − a~ke −ikx]f3(ω2 k, ~k2), (II.9) which is the expected result ∂jϕ(x) because f3 is also
an equivalent partition of unity. For all other commuta-tors the reduction procedure due to the a’s and a+’s is
the usual one and since any power of the test function is an equivalent partition of unity, the Poincar´e algebra is thereby satisfied. It is clear that any arbitrary test function will fail in that respect.
III.
BOGOLIUBOV-SHIRKOV-EPSTEIN-GLASER (BSEG) EXPANSION OF THE
S-MATRIX AND LORENTZ INVARIANT
EXTENSION OF SINGULAR DISTRIBUTIONS A. The BSEG procedure
In the formal expansion of the S−matrix [2] the coeffi-cients are OPVD built out of products of free fields. Ac-cording to the above construction products of test func-tions appear then naturally and in the BSEG writing of S they are made explicit
S(g) = 1+ ∞ X n=1 1 n! Z d4x1..d4xnTn(x1, .., xn)g(x1)..g(xn). (III.1) The test functions g(x) must have two properties:
-i) vanish at infinite time in order to switch off the interaction. This is necessary to define asymptotically free states. It is a problem in equal time QFT because non-trivial vacua are excluded, but in LCQFT, because of the LC-vacuum structure, one can build a non perturbative theory on free field operators.
-ii) restrict the propagation of fields to the interior of the light cone.
Suppose that all time arguments of the set of vec-tors {x1,· · · , xm} are bigger than those of the set
{xm+1,· · · , xn} then the supports of the ensemble of test
functions{g(x1),· · · , g(xm)} are all separated from those
of {g(xm+1,· · · , g(xn)} . The requirement of causality
leads to
Tn(x1,· · · , xn) = Tm(x1,· · · , xm)Tn−m(xm+1,· · · , xn).
(III.2) As already mentioned in the introduction this condition completely fixes the dynamics of the system. If all ar-guments are different and ordered such that x0
1 > x02 >
· · · > x0
n, by recursion it follows that Tn(x1,· · · , xn) =
T1(x1)T1(x2)· · · T1(xn) and S(g) = 1+ ∞ X n=1 1 n! Z d4x 1..d4xnT [T1(x1)..T1(xn)]g(x1)..g(xn). (III.3) It is important to note that the time ordering operation T cannot be made with θ-functions because the multi-plication of the two distributions θ(x) and Ti(x) at the
same point is mathematically ill-defined. Epstein-Glaser analysis focusses then on the solution of this difficulty to which we now turn.
B. Epstein-Glaser extension of singular distributions
Because the functional integration of Eq.(II.1) may oc-cur in practice in a space of smaller dimension than the original dimension D we shall use d as a dimensional la-bel henceforth. Due to translation invariance the multi-plication problem just mentionned can be reduced to the study of distributions singular at the origin of IRd. Let then f (X) be a C∞ test function belonging to D(IRd)
and T (X) a distribution belonging toD′(IR d
\{0}) which we want to extend to the whole domain D′(IRd). The
singular order k of T (X) at the origin of IRd is defined as
k = inf{s : lim
λ→0λ
sT (λX) = 0
} − d. (III.4) Epstein-Glaser original extension consists in performing an ‘educated‘ Taylor surgery on the initial test function by throwing away the weigthed k-jet of f (X) at the ori-gin. Denoting by Rk
implies Rk 1f (X) = f (X)− k X n=0 X |α|=n Xα α! ∂ αf (X) |X=0. (III.5)
Here the notations are : α! = α1!· · · αd!, |α| = α1+
· · · + αd and ∂α= ∂α1
x1 · · · ∂
αd
xd. R
k
1f (X) has the desired
properties at the origin -it vanishes there to order k + 1- but, due to uncontrolled UV-behaviour, it does not belong to Schwartz-space D(IR d). In order to be able to define a functional which can be used to perform a transposition in the sense of distributions Epstein and Glaser introduce a weight function w(X) belonging to D(IR d) with properties w(0) = 1, w(α)(0) = 0 for
0 <| α |≤ k. This allows to define a modified Taylor remainder eg: Rkwf (X) = f (X)− w(X) k X n=0 X |α|=n Xα α! ∂ αf (X) |X=0. (III.6) Now the extension eTw(X) of T (X) can be defined by
transposition:
< eTw, f >:=< T, Rkwf > . (III.7)
The IR-extension eTw(X) so obtained is not unique. We
shall come back to this point . There is an abundant lit-erature on this original procedure of Epstein-Glaser and some important improvements were proposed recently [9, 10]. Our observation here [17] concerns the use of test functions belonging toD(IR d) and vanishing at the origin of IR d with all their derivatives (dubbed super-regular or SRTF hereafter). In this case one has strictly f (X) = Rk
1f (X), for all the derivative terms in Eq.(III.5)
are zero. Then Rk
1f (X) belongs also toD(IR d), there is
no need to introduce a weight function and the following set of equalities holds
< T, f >=< T, Rk1f >=< Rk1T, f >≡< eT , f > . (III.8)
According to Section II f stands here for a partition of unity which allows defining < T, f > from theorem C. Then, eT becomes a regular distribution over the whole manifold on which f is just 1 everywhere. The way this is achieved in practice will be shown in Section IV. Important consequences follow from these relations. They have to do with the splitting procedure of dis-tributions into advanced and retarded parts, Lorentz covariance of the Epstein-Glaser analysis, the connection with BPHZ renormalization and consequently the analy-sis of the IR and UV behaviour of the underlerlying QFT.
C. Splitting into advanced and retarded parts of singular distributions
As mentionned earlier the time ordering operation T in Eq.(III.3) cannot be made blindly without facing
di-vergences related to ill-defined products of distributions. This ordering operation requires to split distributions T into advanced and retarded parts Ta and Tr respectively
in the following way
T (x1,· · · , xn) = Tr(x1,· · · , xn)− Ta(x1,· · · , xn).
(III.9) The support of Tr(x1,· · · , xn) (viz. Ta(x1,· · · , xn) )is
the n-dimensional generalization of the closed foreward (backward) cone of {x1,· · · , xn}. Because of
transla-tional invariance one can put xn = 0. If T (x1,· · · , xn)
were regular at xi= 0, i = 1,· · · , n−1 the splitting could
be performed with Tr(x1, .., xn) = n−1Y i=1 θ(x0j − x0n)T (x1− xn, .., xn−1− xn, 0), Ta(x1, .., xn) = n−1Y i=1 θ(x0n− x0j)T (x1− xn, .., xn−1− xn, 0). (III.10) On the contrary, if the product of θ-functions with the distribution T is ill-defined the splitting procedure has to be done with the help of Eq.(III.8). The defining equa-tions of eTr and eTa are then
< T, θf > = < θ eT , f >=< eTr, f >; (III.11)
< T, (1− θ)f >=< (1 − θ) eT , f >=< eTa, f > .
e
Trand eTaare simply obtained by multiplication of eT with
the corresponding θ-functions. With the original singular distribution T this would not have been possible. Finally the following identification is obtained
e
Tr= θ eT ; Tea= (1− θ) eT . (III.12)
It should be kept in mind that eTrand eTaare not unique,
since, as stated above, eT is not unique.
If T (X) = T (x1,· · · , xn) is causal -as is the case, if
it is constructed with the BSEG procedure sketched in Section (III.A)- then the products of θ-functions in Eq.(III.10) allow an essential simplification partic-ularly useful for calculations in momentum space [4]. There Scharf defines a 4(n − 1)-dimensional vector v = (v1, v2,· · · , vn−1) consisting of n − 1 time-like
four-vectors lying all in the interior of the foreward light-cone. The scalar product vi· Xi = v0iXi0− −→v ·−→X
is used to define the scalar v· X =Pn−1j=1vj· Xj, where
one puts Xn = 0. Then v·X has the following properties:
v· X > 0, ∀Xj in the foreward light-cone Γ+
v· X < 0, ∀Xj in the backward light-cone Γ−
parts.
The following indentities then hold:
θ(v· X) = n−1Y j=1 θ(Xj0− Xn0); X ∈ Γ+ θ(v· X) = n−1Y j=1 θ(Xn0− Xj0); X ∈ Γ− (III.13)
These identities can only be used with distributions having causal supports. Otherwise the scalar products vi· Xiwould not have a unique sign related to advanced
or retarded coordinates and v · X = 0 could not be used to distinguish advanced and retarded parts of the supports. For later use we need the Fourier transform of θ(v· X), i.e. θv(p) where p = (p1, p2,· · · , pn−1).
This multi-dimensional quantity can be reduced to a 4-dimensional one by
(1) choosing a coordinate system for which p = (p1, 0,· · · , 0) where p1 = (p01, −→p1) (which can
be obtained by an orthogonal transformation in
IR4(n−1)).
(2) and choosing v = (v1, 0,· · · , 0); v1= (1, −→v ) which
yields θ(v· X) = θ(X0 1−−→X1· −→v ). Then θv(p) = 1 (2π)D/2 Z dDX 1eip.X1θ(X10−−→X1· −→v ) (III.14) Now with θ(X10−−→X1· −→v ) = 1 2πi Z ∞ −∞ dτe iτ (X10− −→ X1·−→v ) τ− iǫ , one obtains: θv(p) = (2π)(D/2−1) 1 p0+ iǫδ (D−1)(−→p − p0−→v ). (III.15)
The last equality means that −→p and −→v come out paral-lel to each other. Of course physical results should not depend upon the choice of the arbitrary vector v. This will be verified in Section V.
D. Lorentz invariance of Epstein-Glaser procedure and super-regular test functions.
In the Taylor surgery above, under the action of an element Λ of the Lorentz group, derivatives transform as
xα∂α(Λf ) = xα[Λ−1]βα(∂βf )◦ Λ−1,
= (Λ−1X)β(∂βf )◦ Λ−1, (III.16)
and in the general Taylor expansion xα∂
α(Λf )(0) =
(Λ−1X)β∂
βf (0). Lorentz invariance is therefore violated
in this procedure. However, as mentionned earlier, eT (X) is determined up to a sum of derivatives of δ functionsX
|α|≤k
(−1)αaα α!δ
(α)(X). But these δ-terms transform as
δ(α)(ΛX) = [Λ−1X]α
βδ(β)(X). Epstein-Glaser remedy
is then to determine the aα’s to correct for the
viola-tion due to derivatives. With SRTF vanishing at the origin with all their derivatives, because of the identity f (X) ≡ Rk
1f (X), Lorentz violating derivatives are just
not there. Provided then that the space of test functions to be used is restricted to SRTF types, Lorentz invari-ance is satisfied from the outset. Formally eT (X) is only determined up to the sum of derivatives of δ-terms but their contributions are redundant with super-regular test functions. Nevertheless its presence should not be forgot-ten and may prove instrumental in restoring some other broken symmetries [4].
E. BPHZ renormalization: a corollary of Epstein-Glaser procedure.
One of the obstacles for a widespread application of the Epstein-Glaser approach was the non evident link to other renormalization schemes and in particular the question of its multipicative structure and its feasability in renormalization group studies. The chain of identities of Eq.(III.8) valid for SRTF allows to establish quite sim-ply the link of our approach with BPHZ renormalization. In the context of the Ces`aro interpretation of diverging integrals [23] this link has been discussed by Prange [9] and Gracia-Bondia [10]. In the approach with SRTF the link to BPHZ is obtained as follows.
Fourier and inverse Fourier transforms are defined as F(p) = F[f](p) = Z ddX (2π)d/2e −ipXf (X); F(p) = F−1[f ](p) = Z ddX (2π)d/2e ipXf (X). (III.17)
Then, with µ a multi-index as defined after Eq.(III.5), the following relations hold
(Xµf )(p) = (−i)|µ|∂µF(p),
∂µf (0) = (−i)|µ|(2π)−d/2< pµ, F >,
(Xµ)(p) = (−i)|µ|(2π)d/2δ(µ)(p). (III.18)
The Fourier transform of the singular T (X) is well de-fined since the functional built with SRTF identical to their Taylor remainder is itself well defined. Therefore relations (III.8) and (III.18) imply
This is BPHZ substraction at zero momentum, up, as is well known, to an arbitrary polynomial in p originat-ing from the sum of δ-terms mentioned previously. It is known that for a non-massive QFT the BPHZ method faces infrared divergences. However, a mass scale can be introduced by doing substractions at some external momenta q 6= 0. For non-SRTF it amounts to choos-ing the weight function w(X) = eiqX. This is a
legiti-mate choice provided the functional integral in Eq.(III.7) is given a meaning in the sense of Ces`aro summability [10, 23]. However the situation is much simpler with SRTF. One has still eiqXf (X)
≡ Rk
1(eiqXf (X)) and the
chain of equalities in Eq.(III.19) can be rewritten with this modification of f , leading now to BPHZ substrac-tion at momentum p = q. Thus BPHZ appears just as a special case of Epstein-Glaser method. In the next sec-tion we examine in more details UV and IR behaviours when using partition of unity test functions introduced in Section II.
IV. IR AND UV EXTENSIONS OF T (X) WITH
LAGRANGE’S FORMULA FOR TAYLOR’S REMAINDER AND PARTITION OF UNITY
A. Lagrange’s formula in the IR
It is known from general functional analysis that Tay-lor’s remainder can be expressed by Lagrange’s formula.
It is only recently [8] that its use was advocated in the Epstein-Glaser context of extension of singular distribu-tions and in the ensuing RG-analysis. With Lagrange’s formula the transposition operation defining the exten-sion eT (X) reduces to partial integrations in the func-tional integral. According to Lagrange Rk
1f (X) can be written as Rk1f (X) = (k + 1) X |β|=k+1 Xβ β! Z 1 0 dt(1− t)k∂β (tX)f (tX) . (IV.1) For a SRTF∈ D(IR d) it is easily verified that for any k Eq.(IV.1) expresses the identity f (X)≡ Rk
1f (X). Since
this identity is also valid for any power of f (X), the field construction and Poincar´e commutator algebra keep their initial properties with respect to the partition of unity (section II-(A,B)). One has then
< T, f >=< T, Rk1f >= (k + 1) X |β|=k+1 Z ddXT (X)X β β! Z 1 0 dt(1− t)k∂β (tX)f (tX) . (IV.2)
Performing partial integrations in X, surface terms are avoided with SRTF. This is necessary to validate the usual operations with distributions. It turns out from the generic features of f (X) [17], detailed in Appendix B, that the t-integral has an effective lower cut-off at ˜
µkXk < 1 for ˜µ < 1, where kXk < 1 is the norm of X. The extension is looked for in the IR region of the variable X. Since in the following X can be a spatial or
a momentum variable ”infrared” in the present context can mean UV (small distance) or IR (large distance) in physical terminology. Since for a momemtum variable the extension eT<(X) is infrared and/or ultraviolet
regu-lar by construction, the test function can now be replaced by 1 over the whole domain of integration [17]. One ob-tains: e T<(X) = (−)k+1(k + 1) X |β|=k+1 ∂XβX β β! Z 1 ˜ µkXk dt(1− t) k t(k+d+1)T ( X t ) . (IV.3)
For an homogeneous distribution, that is T (X
t) =
e T<(X) = ( −)k(k + 1) X |β|=k+1 ∂XβX β β! T (X) log(˜µkXk) +(−) k k! Hk X |β|=k Cβδ(β)(X), (IV.4) with Hk = k X p=1 (−1)(p+1) p k p = γ + ψ(k + 1) and Cβ=R (kXk=1)T (X)X βdS.
It is interesting to note how this result [24] parallels that of Ref.[10]. Here it is the behaviour of the SRTF at the origin which provides the lower bound in the t-integral and at the same time gives the identity f (X)≡ (1 − w(X))R1k−1f (X) + w(X)Rk1f (X) ∀w(X).
It just corresponds to the Tw operation of [10] applied
on any test function φ(X) ∈ D(IR d) with the choice
w(X) = θ(1− ˜µkXk).
The power of the IR treatment through Eq.(IV.4) is shown in [10, 17] for the free massive scalar field the-ory developed in the (conventionally hopeless) mass per-turbation expansion. It leads through the exact resum-mation of the infinite mass-expansion series to the well known results given in terms of modified Bessel functions Kν(mkXk). Another interesting test concerns integrable
conformal field theories with additive interacting terms of massive dimension, which endure also conventionally untractable IR divergences if treated perturbatively [25].
B. Extension of T (X) in the UV domain
From the above IR analysis one could also explore the Fourier space UV domain. However the same formalism can be directly applied if T (X) gives rise to UV diver-gences in the absence of test functions. In QFT T (X) is in general a distribution in the variablekXk only. Setting kXk ≡ X from now on, the domain of f(X) is the ball B1+h(kXk) of radius 1 + h and f(α)(1 + h) = 0,∀α ≥ 0.
With f (X)≡ f>(X), it also holds that
f>(X)≡ −k!1 Z ∞ 1 dt(1− t)k∂(k+1) t tkf>(Xt), =−Xk! Z ∞ 1 dt t (1− t) k∂(k+1) X Xkf>(Xt).(IV.5)
Whatever the construction of the partition of unity underlying f (X) h is a parameter which may depend on kXk. The arguments underlying this observation are as follows. With h taken as a true constant the regularisation will be achieved in the usual cut-off like manner with well known symmetry preserving problems. In this case the final extension of the partition of unity to the whole manifold can only take place by finally letting h -the cut-off- going to infinity thereby losing the scaling information originally contained in the construction of the overlapping domains. Clearly using a partition of unity would be a useless artifact in this case. However with an kXk-dependant h, as we shall see, the situation is very different: the final extension of the partition of unity to the whole manifold can be achieved altogether in a manner preserving symmetries and scaling informations. With ankXk-dependant h the consequences are:
−i) ∃ kXkmaxsuch thatkXkmax= 1 + h(kXkmax)≡
µ2
kXkmaxg(kXkmax) =⇒ g(kXkmax) = µ12,
−ii) h > 0 =⇒ µ2kXkg(kXk) > 1 ∀ kXk ∈
[1,kXkmax] =⇒ g(1) > g(kXkmax),
−iii) from f>(tX) present in Lagrange’s formula one
has t <1+h(kXk)kXk = µ2g(kXk).
In the definition of h(kXk) a dimensionless scale factor µ2has been extracted from g(kXk). Then one has
< T, f>> = Z ddXT (X)−Xk! Z µ2g(kXk) 1 dt t (1− t) k∂(k+1) X Xkf>(Xt) = < eT>, 1 >, (IV.6)
T (X) over the whole X domain on which f is just one everywhere. . e T>(X) =(−) k k! X (k−d+1)∂(k+1) X XdT (X) Z µ2g(kXk) 1 dt t (1−t) k. (IV.7) A direct application of this relation in the Euclidean met-ric is made for the scalar propagator ∆(x− y) which is diverging for D = 2· · · 4 when x = y. These cases are im-portant to clarify the role of the presently unknown func-tion g(kXk), on the one hand in producing a regular
UV-extension eT>(X) and, on the other hand, in the final
RG-analysis with respect to the scale parameter µ present in f (X). Suppose now that T (X) ∼ kXk−(ω+1) as
kXk → ∞, thenR d(d)XT (X) diverges if (ω + 1
− d) ≤ 0. Thus k in Eq.(IV.7) should be such that k≥ (d − ω − 1). Consider the Euclidean scalar propagator at x = y and for D = 4. The argument of the test function is dimen-sionless. Hence we have X = Λp22, where Λ is an arbitrary
scale which will prove irrelevant. Thus T (X) =XΛ21+m2
and for D = 4 the space dimension in the X-variable is d = 2 and k = 1. We have then
^ h 1 (p2+ m2) i µ,D=4 = −∂ 2 X X2 (XΛ2+ m2) Z µ2g(X) 1 dt(1− t) t . (IV.8)
The full evaluation of Eq.(IV.8) is straightforeward. In keeping with the D = 2 case [17] the elimination of all non self-converging contributions in the X-integral re-quires that g(x) = x(α−1) (up to a multiplicative
arbi-trary constant already taken into account as µ2) and that
the limit α→ 1− is performed. Then h(x) = µ2xα− 1
with 0 < α < 1 is consistent with the generic construc-tion of f (X) (cf Appendix B). It implies also g(1) = 1 >
g(Xmax) = µ12 ie µ2> 1 and Xmax= (µ2)(
1
(1−α)). In the
limit α → 1− the upper integration limit in X extends
then to infinity, the X−integral is self-converging and the test function can be taken to unity over the whole inte-gration domain. In this limit the propagator at x = y becomes ∆(0) = lim α→1 Z d4p (2π)4 f2(p2) (p2+ m2) = 2m 4Z d4p (2π)4 (µ2 − 1 − log(µ2)) (p2+ m2)3 = m 2 (4π)2(µ 2 − 1 − log(µ2)), (IV.9)
which is the expected result with respect to the scale parameter µ [26]. One sees also that this parameter µ will always be present independantly of the infinitely many possible partitions of unity (cf section II) that can be used to build up f (X). In this sense µ is universal, hence its relevance for the final RG-analysis. The fundamental achievement of Lagrange’s formula is that it perfoms the scaling analysis of the distribution T (X) through the integral on the t variable which incorporates the relevant scaling informations carried by µ. The analysis of cases involving loops (two- and four-points vertex functions) follows the same line and is detailed in Appendix C.
Technically things are a bit more involved in Minkowski’s metric. The test function becomes a function ρ(x0, −→x ) in coordinate space or f (p20, −→p2) in
momentum space. As usual it is preferable to first
carry out the integrations over x0 or p0 respectively and
then the remaining integrals. There are two possible situations:
(1) The integrals over x0 or p0 are convergent. In this
case the remaining integrations are as in the Euclidean space with dimension D − 1 and the extension is calculated correspondingly. As is shown in section V there is no divergence problem for the p0-integration in
the calculation of the retarded and advanced extended distributions of causal distributions.
(2) The integrals over x0 or p0 diverge. This requires
an extension of the distribution with respect to the dependance on x0or p0.
case. In principle there can be an UV-divergence prob-lem of the p0-integration or there can be nonintegrable
poles at finite values of p0. The latter singularities are
of IR-nature. Such situations can arise when extending non-causal distributions like powers of Feynman propa-gators. Once the p0-extended distribution has been
cal-culated the remaining integrations are to be done as in (1).
As a matter of illustration we recalculate the (Euclidean) result of eq.(IV.9) in Minkowski’s metric (example of in-tegrable pole contribution). We want to give a meaning to the diverging integral ∆(0) at D = 2, 4
∆(0) = Z dDp f 2(p2 0, p2) p2 0− p2− m2+ 2iǫωp = − Z d(D−1)p 2ωp Z ∞ −∞ dp0[ 1 ωp− p0− iǫ + 1 ωp+ p0− iǫ ]f2(p20, p2) (IV.10)
The p0-integration cannot be done using contour
inte-gration because the extension of the test function to the
whole complex plane is not possible in general. However one can proceed as follows. One has
P P Z ∞ −∞ dp0f 2(p2 0, p2) p0± ωp = lim ǫ→0{ Z ∓ωp−ǫ −∞ dp0+ Z ∞ ∓ωp+ǫ dp0} 1 p0± ωp [−p0 d dp0 Z ∞ 1 dt t f 2(p2 0t2, p2)] = lim ǫ→0[± ωp ǫ − 1 ∓ ωp ǫ ± ωp ǫ + 1∓ ωp ǫ ] log[µ 2] = 0, (IV.11)
where a partial integration on p0 has been performed.
Hence Z ∞ −∞ dp0 f 2(p2 0, p2) p0± ωp∓ iǫ =±iπf 2(ω2 p, p2). (IV.12)
At dimension D = 2 one obtains ∆(0) =−iπ Z ∞ −∞ dp ωp f2(ωp2, p2), (IV.13) =−iπ < 1 ωp , f2(ω2p, p2) >=−iπ < f1 ωp , 1 > .
Here d = 1, ω = 0, and k = 0. Thus g ( 1 ωp ) = d dp[p Z µ2 1 dt t 1 p p2+ m2t2], = p 1 p2+ m2 − 1 p p2+ m2µ4. (IV.14)
The end result for ∆(0) scales then as log[µ2] as expected.
The two contributions in Eq.(IV.14) may be recombined to give a PV-type of substraction at the level of the
prop-agator, as discussed more extensively in the next sub-section ∆(0) = Z d2p f 2(p2 0, p2) p2− m2+ iǫ, (IV.15) = Z d2p[ 1 p2− m2+ iǫ− 1 p2− m2µ4+ iǫ].
It is important to note that causality is restored since the non-causal diverging term in δ(x2) now present in each
contribution of Eq.(IV.15) just cancels out in the sub-straction. Here this is a consequence of ”repairing” an ill defined Feynman propagator. In CPT it is a general feature that - by construction - diverging causality vio-lating terms are avoided right from the beginning [4]. At dimension D = 4 one has
∆(0) = −4iπ2Λ3 Z ∞ 0 XdXf2(X) p X(XΛ2+ m2), = −4iπ2Λ3<p ^1 X(XΛ2+ m2), 1 > .(IV.16)
Here d = 2, ω = 0, and k = 1. Thus
The final t and X integrations give back the result of Eq.(IV.9).
C. Transcription of Epstein-Glaser method in terms of Pauli-Villars type of substractions at the
level of propagators
Again we consider the scalar propagator ∆(x− y) in dimension D = 2 for the simplest demonstration of this transcription. It proceeds as follows.
An alternative form of eT>(X) is obtained through the
change of variables Xt→ Y in the first line of Eq.(IV.6). It gives e T>(X) = (−1) k k! X (k−d+1)∂(k+1) X Xd Z µ2 1 dt(1− t) k t(d+1) T ( X t ) (IV.18)
Setting kXk ≡ X as before, we have, in Euclidean metric for D = 2, d = 1 and k = 0 which gives
^ h 1 (p2+ m2) ialter µ,D=2 = ∂X X Z µ2 1 dt t 1 (XΛ2+ m2t) = 1 (p2+ m2)− 1 (p2+ m2µ2).
This is a Pauli-Villars subtraction, however without any additionnal scalar field. The final momentum in-tegration gives the familiar RG-invariant result ∆(0) =
1 (4π)log(µ
2). The situation for D = 4 is a bit more
intri-cate but also carries usefull informations about the limi-tations of the PV-subtractions with respect to the scaling analysis. As seen above the dimension in the X variable is d = 2 and k = 1. The scalar propagator at x = y takes then a form somewhat different from that given in Eq.(IV.8), ^ h 1 (p2+ m2) ialter µ,D=4 = −∂ 2 X X2 Z µ2 1 dt t2(1− t) 1 (XΛ2+ m2t) (IV.19) = −2 Z µ2 1 dt t2(1− t) 1 XΛ2+ m2t− 2XΛ2 (XΛ2+ m2t)2 + X2Λ2 (XΛ2+ m2t)3 .
It is verified that the final X-integration gives the very same result for ∆(0) as in Eq.(IV.9). In Eq.(IV.19) the expression I(X, m) in brackets is regular at X = 0 and behaves as 1
X3 for X → ∞. Relation (IV.19) is a
particu-lar type of PV substraction. Indeed it may be written as (ignoring t and Λ which are inessential for the argument)
I(X, m) = 1 X + m2 1−αX + β X + Λ2 1 + γX 2+ δX + ǫ (X + Λ2 1)(X + Λ22) = A + BX + CX 2 (X + m2)(X + Λ2 1)(X + Λ22) , (IV.20) with α = 2, γ = 1, β = δ = ǫ = 0 and Λ1 = Λ2 = m.
In the general form of Eq.(IV.20), imposing the fall off in 1
X3 for X → ∞ gives B = 0, C = 0. Without loss of
generality one may take δ = ǫ = 0 then β = Λ2
1+ (1−
α)Λ2
2 , γ = (α− 1) and A = Λ42(α− 1). Normalizing
such that A
(m2−Λ2
1)(m2−Λ22) = 1 determines α to give the
known result that a general regularisation of 1
p2+m2 may
proceed through multiplication by Πni=1
(Λ2i−m
2
)
(p2+Λ2
i) with the
decomposition in terms of PV contributions
1 p2+ m2Π n i=1 (Λ2 i − m2) (p2+ Λ2 i) = 1 p2+ m2− n X j=1 Πni=16=j(Λ2i − m2) Πni=16=j(Λ2i − Λ2j) 1 p2+ Λ2 j . (IV.21)
Under this form it is clear that each individual PV-term treated separetly would have to undergo Epstein-Glaser treatment to exhibit its scaling behaviour in terms of µ. It shows that even though FΛ(p2) = Πni=1
(Λ2
i−m 2)
(p2+Λ2
i) might
be viewed as a special case of test function reducing to unity when{Λi} → ∞ and obeying Lagrange’s formula
Eq.(IV.1) with X = p2it does not have the intrinsic
V. CAUSAL SPLITTING OF DISTRIBUTIONS AND LINK WITH DISPERTION RELATION
TECHNICS
Here the calculations of the retarded extension of a singular distribution is worked out explicitely. The result shows an interesting link to substraction technics known from dispersion relations.
The starting point is the following form of Lagrange’s formula for SRTF’s: φ(X) = (ω + 1)X |β|=ω+1 Xβ β! Z 1 0 dt(1− t) ω t(ω+1) ∂ β (X)φ(Xt) . (V.1) It yields fT<
r (X) after partial integrations and taking into
account the restriction brought about by φ(Xt) in the t−integral (cf Appendix B, Eq.(B.3) and the discussion thereafter): f T< r (X) = (−1)(ω+1) X |β|=ω+1 ∂XβX β β! Z 1 1/µ2 dt(1− t) ω t(d+ω+1)θ( v.X t )T ( X t ) . (V.2)
It corresponds to Eq.(IV.3) but with a less refined lower bound on the t−integral. However, in keeping with the PV-type of substraction (cf Eqs.(IV.14-15)), we shall see below that it provides also the interpretation of the
sub-traction in the dispersion relations. The corresponding Fourier transform -which becomes U V−regulated [27], hence its notation- is:
f T> r (p) = (ω + 1) (2π)d/2 X |β|=ω+1 pβ β! Z 1 1/µ2 dt(1− t) ω t(ω+1) Z dkθv(k)∂pβT (pt− k) . (V.3)
Using Eq.(III.15) for θv(k) this becomes:
f T> r (p01) = i 2π (p0 1)ω+1 ω! Z ∞ −∞ dk0 1 k0 1+ iǫ Z 1 1/µ2 dt(1− t)ω∂(p(ω+1)0 1t) T (p 0 1t− k10, 0,· · · , 0). (V.4)
With the change of variables p0
1t− k01 = k1′0 and after ω + 1 partial integrations in k′0 1 one obtains: f T> r (p01) = i 2π (p0 1)ω+1 ω! (1− µ 2)(ω+1)Z ∞ −∞ dk1′0 T (k′0 1, 0,· · · , 0) (p0 1− k′01µ2+ iǫµ2)(ω+1)(p01− k1′0+ iǫ) , (V.5)
Z 1 1/µ2 dt (1− t) ω (p0 1t− k′01 + iǫ)(ω+2) = (µ 2 − 1)(ω+1) (ω + 1)(p0 1− k′01 + iǫ) 1 (p0 1− k1′0µ2+ iǫµ2)(ω+1) . (V.6)
Finally with the substitution k′0
1 = sp01 in Eq.(V.5) and
returning to the general frame [4] we get:
f T> r (p) = i 2π (µ2− 1)(ω+1) (µ2)(ω+1) Z ∞ −∞ ds T (sp) (s− 1/µ2− iǫ′)(ω+1)(1− s + iǫ′). (V.7)
The retarded, extended distribution fT>
r (p) satisfies
an unsubstracted dispersion relation since fT< r (X) of
Eq.(III.12) is a well-defined regular quantity. On the other hand the original retarded distribution T<
r = θT
is singular. A dispersion relation for this quantity would require ω + 1 substractions. It is interesting to see that the explicit construction of the extended distribution in
terms of the original singular distribution leads to the fac-tor (s− 1/µ2
− iǫ′)(ω+1) in the denominator of Eq.(V.7)
which is characteristic of dispersion relations with ω + 1 substractions, with one important difference: the sub-straction point is not arbitrary but µp2, as shown
here-after. It is the scale µ present in the SRTF which fixes this point. The calculation of the advanced, extended distribution fT>
a (p) follows the same lines with a result
similar to that of Eq.(V.7) with iǫ→ −iǫ. The difference of the retarded and advanced pieces reduces to
f T> r (p)− fTa>(p) = T (p)− ω X n=0 (−1)n n! µ2− 1 µ2 nZ ∞ −∞ dsT (sp)δ(n)(s−µ12), = T (p)− ω X n=0 X |β|=n [p(µµ22−1)]β β! T (β) ( p µ2), (V.8)
which is just the Taylor remainder of eT (p − q) at q = µp2 -( eTq(p) in the notation of [4])- that is the BPHZ
substraction at q = µp2 discussed in section III-E.
VI. NON-PERTURBATIVE LIGHT-FRONT QFT
(LFQFT) AND THE BSEG PROCEDURE
At various places in the text we commented on the necessity of using test functions when quantizing fields in Dirac’s front form [28]. As stated in the introduction the restriction of quantum fields to a lightlike surface does not canonically exist and turns out to be the ma-jor mathematical problem to face first in dealing with LFQFT. The question was initially adressed in [13] and fully discussed in [14]. The outcome is the Fock expan-sion of the field operator written in LC-variables with test
functions taking proper account of both the LC-induced IR singularity at k+ = 0 and of the UV behaviour in
k+. For a scalar field theory the Haag series is well
of equations, could be determined inductively. Thereby overlapping contributions and the renormalization diffi-culties mentioned above could be avoided. An immediate objection is that since the BSEG procedure is perturba-tive in essence it cannot fulfil our non-perturbaperturba-tive goals. However the same objection is also valid for the standard
scheme with coupled equations and it is well known that the non-perturbative aspects of LFQFT are encoded in the constrained dynamics related to the singular nature of the light front Lagrangian. To be specific consider the standard expression for the S-matrix [31]
S = 1 +X
n
Z
(−i)nHint(x
1)θ(t1− t2)Hint(x2)· · · θ(tn−1− tn)Hint(xn)d4x1· · · d4xn, (VI.1)
where Hint(x) is the interaction Hamiltonian and the
time ordering is made with θ-functions, albeit the diver-gences occuring on the light cone when (xi−1− xi)2= 0
[32] and subjects of Epstein-Glaser attention. The light front is defined by ω.x = 0, where ω is an arbitrary four vector such that ω2= 0. The expression of Eq.(VI.1) is
then repesented in terms of the light front time σ = ω.x for, if (xi−1− xi)2 > 0, the signs of ω.(xi−1− xi) and
(ti−1 − ti) are the same provided the time ordering is
treated according to the generalized Epstein-Glaser con-siderations developped above which eliminates light-cone divergences. Hence Hint
ω = Hint, however with
impor-tant caveats. We discuss first the case of interacting scalar fields. Obviously the asymptotic states have to be free bosonic fields ϕ0(x) as defined after Eq.(II.4). Since
the LC-Lagrangian is singular, the quantization has to follow the specific rules for constrained systems [34, 35] . In the present case it means that in addition to the equa-tion of moequa-tion for the particle sector field ϕ(x) there ex-ists a constraint which lives only in the zero mode sector and is of nondynamical nature. If this constraint has a nontrivial solution Ω then the total field Φ(x) is a sum: Φ(x) = ϕ(x) + Ω. The zero mode field carries the infor-mation on nontrivial physics and is the LC-counterpart of a nontrivial vacuum in equal-time quantization. Con-sequently the field appearing in the interaction Hint(x) is
the total field Φ(x). There is, however, a problem which cannot be overlooked: the zero mode Ω following from the constraint can only be obtained approximately and recursively in an iterative procedure, which means that the interaction Hint(x) is a priori known only as far as Ω
is known. This was the bad news, shared with alterna-tive formulations such as that of Ref.[30] or the covariant light front dynamics approach of Ref.[29]; the good news is that, if the coupling is large enough to support a zero mode or in other words to generate a phase transition -already the first, algebraically very simple iteration step leads to a form for the zero mode giving a value for the critical coupling in good agreement with conventional, much more complex calculations and permitting to dis-cuss the critical behaviour near the phase transition [16]. Thus there is a good reason to hope that a small num-ber of iteration steps would lead to a proper description
of non-perturbative physics. The presence of zero modes in the field operator does not spoil the causality issues of the theory: causality expressed through the Pauli-Jordan commutator function makes a statement about the mea-surability of fields at different space-time points. Since such a statement concerns only the particle sector of the theory, the zero mode contribution is not involved in the causal commutator.
In fermionic theories the particle sector itself is split into a sum of independent and dependent degrees of freedom which leads -in gauge theories- to additional interactions. The zero mode problem concerns the gauge field itself and interaction terms built out of primary fermionic fields and behaving like scalar fields.
The validation of these ideas in LFQFT gauge theories are undoubtly challenging. Already Yang-Mills theories have been investigated in some details within the stan-dard CPT framework [36], paving the way to the imple-mentation of the CPT-LFQFT studies. They will be of considerable theoretical and even practical interests since the outcome is a rigourous non-perturbative alternative to lattice calculations, enjoying the numerous benefits of an LFQFT formulation put foreward for decades [37].
VII. CONCLUSIONS
re-mainder of any order is identical to the test function it-self. The flexibility in the construction of the test func-tions allows to treat IR- and UV-divergent distribufunc-tions with the same mathematical techniques. By the very na-ture of the construction an arbitrary scale, characterising the building blocks of the decomposition of unity, comes naturally into the picture. Such a scale is the corner stone permitting the use of extended distributions in renormal-ization group studies. Another important aspect of the decomposition of unity concept for the construction of the test functions is that it does not violate gauge invari-ance - contrary to other test function methods. A first treatment [38] in the QED context verifies this for the po-larisation tensor in dimension D = 2 and gives Fujikawa’s analysis of the axial anomaly in dimensions D = 2, 4 di-rectly from the presence of the test function as partiton of unity. A general analysis of gauge invariance is given in [4] and should be further developped in the present context. In coordinate space the extended distribution
differs from the original singular distribution only on the support of the singularity whereas in momentum space it differs from the original one by subtractions at points determined by the intrinsic scale of the test function. Finally we presented arguments, based on the recent developments of Epstein-Glaser causal approach, which make it extremely plausible that a finite symmetry-preserving LCQFT could be envisaged on the basis of an iterative construction of the S-matrix and a causality conditioned finite regularization using the OPVD treat-ments of fields.
Acknowledgements E. Werner is grateful to Alain Fal-vard for his kind hospitality at LPTA and to the IN2P3-Department from CNRS for financial support. We are particularly indebted to M. Slupinski for his expertise in topological analysis. Discussions with M. Weinstein, V. Braun, M. Lavelle, L. Martinovic and P. Ullrich are gratefully acknowledged.
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Appendix
APPENDIX A: COMPOSITION OF THE PRODUCT OF TWO PARTITIONS OF UNITY
The purpose here is to give a proof of the Remark of section II, quoted quite frequently in the literature. Most of the time the proof is left to the reader but since the property is basic to section II it is useful to establish its validity.
Property: If (αi)i∈I and (βj)j∈J are two partitions of
unity subordinate to (Uk)k∈K and (Vl)l∈L respectively,
(αiβj)(i,j)∈I×J is a partition of unity subordinate to
(Uk∩ Vl)(i,j)∈K×L.
1) Clearly (Uk∩ Vl)(i,j)∈K×L is a covering ofM
2) For X∈ M one chooses a neighbourhood UX (resp.
VX) of X such all but a finite number of αi (resp. βj)
are zero on UX (resp. VX). Then all but a finite number
of the products αiβj are zero on the neighbourhood
UX∩ VX de X. This prooves condition d(i) of section II
for the family of functions (αiβj)(i,j)∈I×J.
3) Let α1, . . . , αA et β1, . . . , βB be the αi (resp. βj
which does not vanish on UX (resp. VX). One has then
for all x∈ UX,
α1(x) + . . . αA(x) = 1
and for all x∈ VX,
β1(x) + . . . βA(x) = 1
which implies that for all x∈ UX∩ VX,
(α1(x) + . . . αA(x))(β1(x) + . . . βA(x)) = 1.
The products αiβjappearing in this equation are the only
one which does not vanish on UX∩VX, which implies that
for all x∈ UX∩ VX, one has
X
(i,j)∈I×J
αi(x)βj(x) = 1.
This proves condition d(ii)of section II for the family of functions (αiβj)(i,j)∈I×J.
4) Since the support of a product of fonction is the intersection of the supports, if for i ∈ I on chooses f (i) ∈ K such that suppαi ⊆ Uf (i) and for j ∈ J on
chooses g(j)∈ L such that suppβj ⊆ Vg(j) it will hold
that supp(αiβj)⊆ Uf (i)∩ Vg(j).
This proves condition d(iii)of section II for the family of functions (αiβj)(i,j)∈I×J.
As a consequence the definition of the field given in Eq.(II-5) with f as a partition of unity is independant of the way it is constructed since
X i∈I γi≡ X (i,j)∈I×J αiβj= X (j,i)∈J×I βjαi≡ X j∈J νj .
APPENDIX B: TEST FUNCTIONS AS PARTITION OF UNITY: CONSTRUCTION AND
PROPERTIES.
Since in the analysis of Section IV the generic test function depends only on one variable, the dimension-less variablekXk, here we only need to detail the sim-plest case of the decomposition of unity on the line. The family of functions{βj} building up unity are based on
point x∈ [0, h] u(x) + u(h − x) = 1. This is the sim-plest realisation of the local finiteness property of the {uj} family. By translation for any j ∈ I and for any
x∈ [jh, (j +1)h] one has u(x−jh)+u((j +1)h−x)) = 1.
Then βj(x) = u(x− jh) and P∞j=−∞βj(x) = 1. The
overlaping subsets (Oj) covering the line are visualised
in Fig.(B1).
2h
4h
6h
x
1
beta
jj= 1
j= 2
j= 3
j= 4
j= 5
Fig.B1: The functions βj(x) = u(x− jh) decomposing unity.
Each individual βj vanishes with all its derivatives
out-side (Oj). Due to the independence of the OPVD
func-tional on the construction of the partition of unity there is no need to consider more elaborate constructions in terms of the constituants functions{uj}. From the
rela-tion u(x− jh) + u((j + 1)h − x)) = 1 for x ∈ [jh, (j + 1)h] it follows also that
γj(k) = Z (j+1)h (j−1)h dx u(x− jh) e−ikx (B.1) = 0 if k = 2nπ h , h if k = 0, ∀j. This orthogonality property leads to the Fourier mode analysis in the distributional context. It is also interest-ing to note that the orthogonal Bloch functions approach used in the noncompact lattice formulation of gauge the-ories of Ref.[39] can be then rephrased in terms of the decompositon of unity, the lump functions there being particular realisations of the u(x− jh) with only a fi-nite number of derivatives vanishing outside (Oj). For
completness we give explicitely two constructions of the elementary function u. They are based on the regularis-ing function of L. Schwartz [18]
ρ(x) =N ex2−11 if |x| < 1, 0 if |x| ≥ 1, (B.2)
whereN stems from normalising ρ(x) to unity. u(j,h)ǫ (x)
is then obtained from the convolution of the
character-istic function{χj,h(x) = 1 if jh ≤ x ≤ (j + 1)h, 0
elsewhere}, with the function ρǫ(x) such that ρǫ(x) = 1 ǫρ( x ǫ). One has u (j,h) ǫ (x) 6= 0 for x ∈ [jh − ǫ, (j + 1)h + ǫ] and u(j,h)ǫ (x) = 1 for x∈ [jh + ǫ, (j + 1)h − ǫ]. ThenPN −1j=0 u (j,h) ǫ (x) is a partition of unity on Ω = [0, L]
with L = N h. Another useful construction is achieved, for any ν > 0, with
u(x− jh) = NR|x−jh|h dv exp[− h2ν vν(h−v)ν], for (j− 1)h < x < (j + 1)h; 0 for |x − jh| ≥ h, (B.3) whereN = R0hdv exp[− h2ν
vν(h−v)ν]. It is verified that for
any x∈ [jh, (j + 1)h] one has indeed u(x − jh) + u((j + 1)h−x)) = 1 and that βj vanishes with all its derivatives
outside (Oj) = [(j− 1)h, (j + 1)h].
In the scaled variablekXk the final test function f(X) ≡ f>(X) built in this way is then
1
3
5
X
1
f
1
2
3
4
X
1
f
Fig.B2: The function f (X) from Eqs.(B4) and from u(x− jh) with ν = 1 a)left curves α = 0.95, dashed: µ2= 1.25; full line: µ2= 1.15;
b)right curves µ2= 1.15, dashed: α = 0.95; full line: α = 0.85;
Hence the function χ(kXk, h) is just the very end part of the elementary function u building up unity, say u(kXk − 1) in the last example. Since f(X) vanishes with all its derivatives at kXk = 1 + h, Lagrange’s formula, Eq.(IV.5), is identically fulfilled. Moreover from its construction f (Xt) is different from zero if kXkt < (1 + h). Hence the upper limit of the t-integral in Eq.(IV.5) is tmax = (1+h)kXk . Due to the
regularising exponential at the upper boundary it is easily seen on the two constructions presented above that f (X) still vanishes at an kXk-dependant upper boundary with all its derivatives and faster than any inverse power of kXk. This property is assumed to be preserved for the generic function χ(kXk, h(kXk)). One writes now h(X) = µ2
kXkg(kXk) − 1 where µ is an ar-bitrary dimensionless parameter extracted from g(kXk). It controls then the size of the overlapping regions for two consecutives (Oj).This is shown in Fig.(B2),where
f (X), constructed from Eqs.(B4) and from u(x− jh) with h(X) = µ2
kXkα
− 1, is shown respectively at fixed α(viz .µ2) for two values of µ2(viz .α). With the scale in
the X variable used in Fig.(B2) for 0 <kXk ≤ h f (X) rises so sharply to 1 that it is not distinguishable from a θ−function.
From the discussion in Section IV µ > 1 and
tmax = (1+h)kXk = µ2kXk(α−1) > 1. The maximum
value of kXk is (µ2)( 1
1−α) and tends to infinity when
α tends to 1. For a distribution T (X) singular at the origin of IRd and with the representation of Eq.(B.3) the test function f<(X) behaves simply
as w(X) = u(h − kXk) for kXk ≤ h. Then in the limit h → 0 w(X) tends to a step function θ(kXk − h). When α → 1 h effectively tends to zero for values of kXk of the order of 1
µ2 and then
w(Xt) = θ(kXkt − h(kXkt )) = θ(t − kXk(µ2 − 1)).
Since the extension of T (X) is seeked in the IR and ˜
µ = (µ2
− 1) > 0, ˜µkXk < 1 is a lower bound to the t-integral in Eq.(IV.3).
APPENDIX C: EVALUATION OF
LOOP-CONTRIBUTIONS WITH PARTITION OF UNITY TEST FUNCTIONS.
We consider first the one-loop contribution I(k2) to the
four-point function of Φ4scalar field theory at D = 4. We
shall work in Euclidean metric to simplify the derivation. I(k2) reads I(k2) = Z d4p (2π)4 f2(p2)f2((p + k)2) (p2+ m2)((p + k)2+ m2), (C.1) = Z 1 0 dx Z d4q (2π)4 f2((q + xk)2))f2((q − (1 − x)k)2) (q2+ k2x(1− x) + m2)2 .
The two cases k2= 0 and k26= 0 must be distinguished.
In the first case, with X = Λp22, one has
I(0) = Λ 4 (4π)2 Z ∞ 0 XdXf4(X) (XΛ2+ m2)2. (C.2) With T (X) = (XΛ2X+m2)2, lim X→∞T (X)∼ 1 X. Hence d = 1, ω = 0, k = 0 and then e T (X) = ∂X[ X2 (XΛ2+ m2)2] = 2Xm2 (XΛ2+ m2)3 (C.3) It follows that I(0) =2m 2Λ4 (4π)2 Z ∞ 0 XdX (XΛ2+ m2)3 Z µ2 1 dt t = 1 (4π)2log(µ 2) (C.4) In the second case, k26= 0, since the test function is unity for small values of its argument, in the UV regime xkkk and (1− x)kkk can be disregarded with respect to kqk and the regulation of I(k2) is provided by f4(q2) only
